On the hereditary character of certain spectral properties and some applications

• Carpintero, Carlos Rafael ^[3] ; Rosas Rodriguez, Ennis Rafael ^[1] ; García Mojica, Orlando J. ^[3] ; Sanabria, José Eduardo ^[2] ; Malaver, Andrés
  1. [1] Universidad de la Costa

     Universidad de la Costa

     Colombia

     
  2. [2] Universidad de Sucre

     Universidad de Sucre

     Colombia

     
  3. [3] Corporación Universitaria del Caribe-CECAR.

  Mostrar afiliaciones +
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 40, Nº. 5, 2021, págs. 1053-1069
• Idioma: inglés
• DOI: 10.22199/issn.0717-6279-3678
• Enlaces
  • Texto completo
• Resumen
  • español

    En este artículo estudiamos el comportamiento de ciertas propiedades espectrales de un operador  $T$ sobre un subespacio propio cerrado cerrado y $T$-invariante $W\subseteq X$ tal que $T^n(X)\subseteq W$, con $n\geq 1$, donde $T\in L(X)$ y $X$ es un espacio de Banach complejo infinito-dimensional. Probamos que para estos subespacios un gran número de propiedades espectrales son transmitidas de $T$ a su restricción sobre $W$ y viceversa. Como consecuencia de nuestros resultados, damos damos condiciones para las cuales propiedades de se semi-Fredholm, así como Teoremas tipo Weyl, son equivalentes para dos operados dados. Adicionalmente, damos condiciones bajo las cuales un operador que actúa sobre un subespacio puede ser extendido a todo el espacio preservando los Teoremas tipo Weyl. En particular, damos algunas aplicaciones de estos resultados para operadores integrales que actúan sobre ciertos espacios de funciones.

  • English

    In this paper we study the behavior of certain spectral properties of an operator T on a proper closed and T-invariant subspace W ⊆ X such that Tn (X) ⊆ W, for some n ≥ 1, where T ∈ L(X) and X is an infinite-dimensional complex Banach space. We prove that for these subspaces a large number of spectral properties are transmitted from T to its restriction on W and vice-versa. As consequence of our results, we give conditions for which semiFredholm spectral properties, as well as Weyl type theorems, are equivalent for two given operators. Additionally, we give conditions under which an operator acting on a subspace can be extended on the entire space preserving the Weyl type theorems. In particular, we give some applications of these results for integral operators acting on certain functions spaces.

     

• Referencias bibliográficas
  • P. Aiena, Fredholm and local spectral theory, with application to multipliers. Dordrecht: Springer, 2004.
  • P. Aiena, “Quasi-Fredholm operators and localized SVEP”, Acta scientiarum mathematicarum, vol. 73, no. 1, pp. 251-263, 2007.
  • P. Aiena, M. T. Biondi, and C. Carpintero, “On Drazin invertibility”, Proceeding of the American Mathematical Society, vol. 136, no. 8, pp....
  • F. Astudillo-Villalba and J. Ramos-Fernández, “Multiplication operators on the space of functions of bounded variation”, Demonstratio mathematica,...
  • B. Barnes, “The spectral and Fredholm theory of extensions of bounded linear operators”, Proceeding of the American Mathematical Society,...
  • B. Barnes, “Restrictions of bounded linear operators: closed range”, Proceeding of the American Mathematical Society, vol. 135, no. 6, pp....
  • M. Berkani, “Restriction of an operator to the range of its powers”, Studia mathematica, vol. 140, no. 2, pp. 163-175, 2000.
  • M. Berkani, “On a class of quasi-Fredholm operators”, Integral equations and operator theory, vol. 34, no. 1, pp. 244-249, 1999.
  • M. Berkani and M. Sarih, “On semi B-Fredholm operators”, Glasgow mathematical journal, vol. 43, no. 3, pp. 457-465, 2001.
  • M. Berkani and H. Zariouh, “Extended Weyl type theorems”, Mathematica bohemica, vol. 134, no. 4, pp. 369-378, 2009.
  • M. Berkani and J. Koliha, “Weyl type theorems for bounded linear operators”, Acta scientiarum mathematicarum, vol. 69, no. 1-2, pp. 359-376,...
  • M. Berkani and H. Zariouh, “New extended Weyl type theorems”, Matematički vesnik, vol. 62, no. 2, pp. 145-154, 2010.
  • M. Berkani, M. Sarih, and H. Zariouh, “Browder-type theorems and SVEP”, Mediterranean journal of mathematics, vol. 8, pp. 399-409, 2011.
  • C. Carpintero, A. Gutiérrez, E. Rosas, y J. Sanabria, “A note on preservation of generalized Fredholm spectra in Berkani’s sense”, Filomat,...
  • L. A. Coburn, “Weyl’s theorem for nonnormal operators”, Michigan mathematical journal, vol. 13, no. 3, pp. 285-288, 1966.
  • L. Chen and W. Su, “A note on Weyl-type theorems and restrictions”, Annals of functional analysis, vol. 8, no. 2, pp. 190-198, 2017.
  • J. K. Finch, “The single valued extension property on a Banach space”, Pacific journal of mathematics, vol. 58, no. 1, pp. 61-69, 1975.
  • A. Gupta and K. Mamtani, “Weyl-type theorems for restrictions of closed linear unbounded operators”, Acta Universitatis Matthiae Belii. Series...
  • R. E. Harte and W. Y. Lee, “Another note on Weyl’s theorem”, Transactions of the American Mathematical Society, vol. 349, no. 5, pp. 2115-2124,...
  • H. Heuser, Functional analysis. New York, NY: Marcel Dekker, 1982.
  • E. Hewitt and K. A. Ross, Abstract harmonic analysis, vol. 1. Berlin: Springer, 1963.
  • K. Jörgens, Linear integral operator. Boston, MA: Pitman, 1982.
  • V. Rakočević, “Operators obeying a-Weyl’s theorem”, Revue Roumaine de Mathématique Pures et Appliquées, vol. 34, no. 10, pp. 915-919, 1989.
  • V. Rakočević, “On a class of operators”, Matematički vesnik, vol. 37, no. 4, pp. 423-425, 1985.
  • J. Sanabria, C. Carpintero, E. Rosas, and O. García, “On generalized property (v) for bounded linear operators”, Studia mathematica, vol....
  • H. Zariouh, “Property (gz) for bounded linear operators”, Matematički vesnik, vol. 65, no. 1, pp. 94-103, 2013. [Online]. Available: https://bit.ly/3iVQht4
  • H. Zariouh, “New version of property (az)”, Matematički vesnik, vol. 66, no. 3, pp. 317-322, 2014.
  • H. Weyl, “Über beschränkte quadratische formen, deren differenz vollstetig ist”, Rendiconti del Circolo Matematico di Palermo, vol. 27, pp....